To find the inverse of an element \( x \) in the context of real numbers \( \mathbb{R} \), we first need to clarify the type of operation we're discussing, as "inverse" can have different meanings depending on the type of operation (addition, multiplication, etc.).
1. Multiplicative Inverse
The multiplicative inverse of a number \( x \) (where \( x \neq 0 \)) is defined as the number that, when multiplied by \( x \), yields 1. Mathematically, it is expressed as:
\[ x^{-1} = \frac{1}{x} \]
For example, the multiplicative inverse of \( 5 \) is:
\[ 5^{-1} = \frac{1}{5} \]
2. Additive Inverse
The additive inverse of a number \( x \) is defined as the number that, when added to \( x \), yields 0. Mathematically, it is expressed as:
\[ -x \]
For example, the additive inverse of \( 5 \) is:
\[ -5 \]
Summary
- Multiplicative Inverse of \( x \): \(\frac{1}{x}\) (for \( x \neq 0 \))
- Additive Inverse of \( x \): \(-x\)
If you have a specific operation in mind or a specific context (like matrices or groups), please provide more details for a tailored explanation!